2.29 Numerical Fluid Mechanics Fall 2011 Lecture 3

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1 Numerical Fluid Mechaics Fall 2011 Lecture 3 REVIEW Lectures 1-2 Approximatio ad roud-off errors ˆx a xˆ Absolute ad relative errors: E a xˆ a ˆx, a ˆx a xˆ Iterative schemes ad stop criterio: ˆx 1 a ˆx 1 For digits: 10 s 2 Number represetatios Floatig-Poit represetatio: x = m b e Quatizig errors ad their cosequeces Arithmetic operatios (e.g. radd.m) Errors of arithmetic/umerical operatios Large umber of additio/subtractios, large & small umbers, ier products, etc Recursio algorithms (Hero, Horer s scheme, Bessel fuctios, etc): Order of computatios matter Roud-off error growth ad (i)-stability s Numerical Fluid Mechaics PFJL Lecture 3, 1 1

2 Numerical Fluid Mechaics Fall 2011 Lecture 3: Outlie Trucatio Errors, Taylor Series ad Error Aalysis Taylor series: f ( x i 1 ) f ( x i ) x f '( x i ) x 1 ( 1 R f ) ( ) 1! x 2 x 3 x f ''( x i ) f '''(x i )... 2! 3!! f (x i ) R Use of Taylor Series to derive fiite differece schemes (first-order Euler scheme ad forward, backward ad cetered differeces) Geeral error propagatio formulas ad error estimatio, with examples Cosider y f ( x 1, x 2, x 3,..., x ). I f i' s are magitudes of errors o x i 's, what is the error o y? The Differetial Formula: f( x 1,..., x ) y i i 1 x i f 2 The Stadard Error (statistical formula): 2 E( s y) i i 1 x i Error cacellatio (e.g. subtractio of errors of the same sig) Coditio umber: x f '( x ) K p f ( x ) Well-coditioed problems vs. well-coditioed algorithms Referece: Chapra ad Caale, Numerical stability Chaps 3, 4 ad 5 Numerical Fluid Mechaics PFJL Lecture 3, 2 2

3 Numerical Fluid Mechaics: Lectures 3 ad 4 Outlie Coditio Numbers Roots of oliear equatios Chaps 3, 4 ad 5 Bracketig Methods Example: Hero s formula Bisectio False Positio Ope Methods Ope-poit Iteratio (Geeral method or Picard Iteratio) Examples Covergece Criteria Order of Covergece Newto-Raphso Covergece speed ad examples Secat Method Examples Covergece ad efficiecy Extesio of Newto-Raphso to systems of oliear equatios Roots of Polyomial (all real/complex roots) Ope methods (applicatios of the above for complex umbers) Special Methods (e.g. Muller s ad Bairstow s methods) Referece: Chapra ad Caale, Numerical Fluid Mechaics PFJL Lecture 3, 3 3

4 Trucatio Errors, Taylor Series ad Error Aalysis Taylor Series: Provides a mea to predict a fuctio at oe poit i terms of its values ad derivatives at aother poit (i the form of a polyomial) Hece, ay smooth fuctios ca be approximated by a polyomial Taylor Series (Mea for itegrals theorems): 2 3 x x x i 1 i x f '(x i ) f ''(x i ) f '''(x i )... f (x i ) R f (x ) f (x ) 1 x ( 1) R f ( ) 1! = costat + lie + parabola + etc 2! 3!! Numerical Fluid Mechaics PFJL Lecture 3, 4 4

5 Derivatio of Geeral Differetial Error Propagatio Formula Uivariate Case Recall: Hece, f i 1 ( x ) f ( x ) x f R 1 x 1! i f ( 1) '( x i ) x 2! 1 ( ) O ( x ) y f f( x i 1) f( x i ) For xf'( x ) x 1, f x i x 2! f 2 '( x i ) 2 f ''( x i ) f ''( x i ) x 3! 3 x 3! 2 O ( x ) x f '( x ) 3 f '''( x i )... f '''( x i )... i x! x! f ( x ) R i f ( x i ) R Thus, for a error o x equal to Δx such that x 1, we have a error o y equal to : y y f x f '(x i ) x f '( x i ) f '( x i ) Multivariate case y f (x, x, x,..., x ) f( x 1,..., x ) For x i 1, y i 1 x i i Derivatio doe i class o the board Numerical Fluid Mechaics PFJL Lecture 3, 5 5

6 Geeral Error Propagatio Formula (The Differetial Formula) For this large plotted x, secod derivative ot egligible Absolute Errors? Fuctio of oe variable x = x -x y~f (x) x Geeral Error Propagatio Formula x x Numerical Fluid Mechaics PFJL Lecture 3, 6 6

7 Error Propagatio Example with Differetial Approach: Multiplicatios Multiplicatio Error Propagatio Formula => => => ε r y ε r i => Relative Errors Add for Multiplicatio Aother example, more geeral case: ε r y ε r i Numerical Fluid Mechaics PFJL Lecture 3, 7 7

8 Statistical Error Propagatio Estimate: Error Expectatios ad the Stadard Error Example: Additio a. Choppig Error Expectatio (o average, half the iterval is chopped) b. Roudig Numerical Fluid Mechaics PFJL Lecture 3, 8 8

9 Statistical Error Propagatio Estimate: Error Expectatios ad the Stadard Error y y( x 1, x 2, x 3,..., x ) Stadard Error (assumig Gaussia errors o x i, compute error stadard deviatio o y) 2 Example: Additio Stadard Error is a more accurate measure of expected errors: It is likely that the error will be of that magitude. However, the geeral error propagatio formula is a upper boud Numerical Fluid Mechaics PFJL Lecture 3, 9 9

10 Error Propagatio: Error Cacellatio (occurs whe errors of the same sig are substracted) Example: Cosider this fuctio of oe variable Defie: For 2 terms (z 1 ad z 2 ): Ge. error (max): Stad. Error : Direct error (sigle y term): Error cacellatio 2 2 error if 4 digits Numerical Fluid Mechaics PFJL Lecture 3, 10 10

11 The Coditio Number The coditio of a mathematical problem relates to its sesitivity to chages i its iput values A computatio is umerically ustable if the ucertaity of the iput values are magified by the umerical method Cosiderig x ad f(x), the coditio umber is the ratio of the relative error i f(x) to that i x. Usig first-order Taylor series f ( x ) f ( x ) f '( x)( x x ) Relative error i f(x): Relative error i x: (x x ) x f ( x ) f ( x ) f '( x)( x x ) f( x) f ( x ) Coditio Nb = Ratio of relative errors: K p x f '( x) f ( x ) Numerical Fluid Mechaics PFJL Lecture 3, 11 11

12 Error Propagatio Coditio Number: aother derivatio ad a example y = f(x) x x y y x x (1 ) y y (1 ) f ( x ) f( x)(1 ) Problem Coditio Number Problem ill-coditioed Error cacellatio example With first-order Taylor series Well-coditioed problem Numerical Fluid Mechaics PFJL Lecture 3, 12 12

13 Error Propagatio Problem vs. Algorithm Coditio Numbers a. Problem Coditio Number b. Algorithm (4 Sigificat Digits) Coditio Number 2 K A is the algorithm coditio umber, which may be much larger tha K P due to digitized umber represetatio. Solutio: Higher precisio Rewrite algorithm Well-coditioed Algorithm (base 10) Algorithm Coditio Number Numerical Fluid Mechaics PFJL Lecture 3, 13 13

14 Roots of Noliear Equatios Fidig roots of equatios (x f (x) = 0) ubiquitous i egieerig problems, icludig umerical fluid dyamics applicatios I geeral: fidig roots of fuctios (implicit i ukow parameters/roots) Fluid Applicatio Examples Implicit itegratio schemes Statioary poits i fluid equatios, stability characterizatios, etc Fluid dyamical system properties Fluid egieerig system desig Most realistic root fidig problems require umerical methods Two types of root problems/methods: 1.Determie the real roots of algebraic/trascedetal equatios (these methods usually eed some sort of prior kowledge o a root locatio) 2.Determie all real ad complex roots of polyomials (all roots at oce) Numerical Fluid Mechaics PFJL Lecture 3, 14 14

15 Roots of Noliear Equatios Bracketig Methods Bracket root, systematically reduce width of bracket, track error for covergece Methods: Bisectio ad False-Positio (Regula Falsi) Ope Methods Systematic Trial ad Error schemes, do t require a bracket (ca start from a sigle value) Computatioally efficiet, but do t always coverge Methods: Ope-poit Iteratio (Geeral method or Picard Iteratio), Newto-Raphso, Secat Method Extesio of Newto-Raphso to systems of oliear equatios Roots of Polyomials (all real/complex roots) Ope methods (applicatios of the above for complex umbers) Special Methods (e.g. Muller s ad Bairstow s methods) Numerical Fluid Mechaics PFJL Lecture 3, 15 15

16 Bracketig Methods: Graphical Approach A fuctio typically chages sig at the viciity of the root bracket the root f(x) f(x) f(x) f(x) x x x x x t x a x t x a x t x a xt xa (i) (ii) (i) (ii) f(x) f(x) x x x t x a x t x a (iii) (iv) These graphs show geeral ways i which roots may occur o a bouded iterval. I (i) ad (ii), if the fuctio is positive at both boudaries, there will be either a eve umber of roots or o roots; i the other two graphs, if the fuctio has opposite sigs at the boudary poits, there will be a odd umber of roots withi the iterval. Image by MIT OpeCourseWare. These graphs show some exceptios to the geeral cases illustrated o the left. I (i), a multiple root occurs where the fuctio is tagetial to the x axis. There are a eve umber of itersectios for the iterval. I (ii) is depicted a discotiuous fuctio with a eve umber of roots ad ed poits of opposite sigs. Fidig the roots i such cases requires special strategies. Numerical Fluid Mechaics PFJL Lecture 3, 16 16

17 Roots of Noliear Equatios: Bracketig Methods If Example Square root Hero s Priciple Guess root a=2; =6; hero.m g=2; % Number of Digits dig=5; sq(1)=g; for i=2: sq(i)= 0.5*radd(sq(i-1),a/sq(i-1),dig); ed ' i value ' [ [1:]' sq'] hold off plot([0 ],[sqrt(a) sqrt(a)],'b') hold o plot(sq,'r') plot(a./sq,'r-.') plot((sq-sqrt(a))/sqrt(a),'g') leged('sqrt','x','s/x','relative Err') grid o If i value Mea is better guess ( ) / 2 Iteratio Formula ( ) / Numerical Fluid Mechaics PFJL Lecture 3, 17 17

18 Roots of Noliear Equatios: Bracketig Methods Bisectio Icremetal search method defied as: Keep dividig i two the iterval withi which at least oe root lies Algorithm = +1 f(x) yes x o Numerical Fluid Mechaics PFJL Lecture 3, 18 18

19 Roots of Noliear Equatios: Bracketig Methods Bisectio Algorithm = +1 % Root fidig by bi-sectio f=ilie(' a*x -1','x','a'); a=2 bisect.m figure(1); clf; hold o x=[0 1.5]; eps=1e-3; err=max(abs(x(1)-x(2)),abs(f(x(1),a)-f(x(2),a))); while (err>eps & f(x(1),a)*f(x(2),a) <= 0) xo=x; x=[xo(1) 0.5*(xo(1)+xo(2))]; if ( f(x(1),a)*f(x(2),a) > 0 ) x=[0.5*(xo(1)+xo(2)) xo(2)] ed x err=max(abs(x(1)-x(2)),abs(f(x(1),a)-f(x(2),a))); b=plot(x,f(x,a),'.b'); set(b,'markersize',20); grid o; ed Stop criterio above based o x~0 ad f(x)~0 yes o Numerical Fluid Mechaics PFJL Lecture 3, 19 19

20 Roots of Noliear Equatios: Bracketig Methods Bisectio Stop criteria: whe relative estimated error or relative added value is egligible: a 1 xˆ r xˆ r s xˆ r Each successive iteratio halves the maximum error, a geeral formula thus relates the error ad the umber of iteratios: x 0 log 2 E ad, where x 0 is a iitial error ad E ad the desired error, Numerical Fluid Mechaics PFJL Lecture 3, 20 20

21 Aother Bracketig Method: The False Positio Method (Regula Falsi) Bisectio is a brute force scheme Somewhat iefficiet approach Does t accout for magitudes of f(x L ) ad f(x U ) f( xu ) x x x ( L x U ) r U f( xl ) f( x U ) False Positio f( x ) f( x ) f( x ) f( x ) x x ) 0 U x A graphical U x L depictio of the false positio method, Regula U Falsi. Shaded i gree are the r L r U triagles used to derive the method. U L r U ( r f( xl ) f( x ) (x x ) (x x ) f(x) x r f(x u ) x L x u x f( xu ) x x x ( L x U ) r U f( x ) f( x ) L U f(x L ) Image by MIT OpeCourseWare. Numerical Fluid Mechaics PFJL Lecture 3, 21 21

22 Commets o Bracketig Methods (Bisectio ad False Postio) Bisectio ad False Positio: Coverget methods, but oe eeds to miimize fuctio evaluatios Error of False Positio ca decrease much faster tha that of Bisectio, but if the fuctio is far from a straight lie, False Positio will be slow (oe of the ed poit remais fixed) Remedy: Modified False Positio e.g. if ed poit remais fixed for a few iteratios, reduce ed-poit fuctio value at ed poit, e.g. divide iterval by 2 Oe ca ot really geeralize for Root Fidig methods (results are fuctio deped) Always substitute estimate of root x r at the ed to check how close estimate is to f(x r )=0 Numerical Fluid Mechaics PFJL Lecture 3, 22 22

23 Roots of Noliear Equatios: Ope Methods Fixed Poit Iteratio (Geeral Method or Picard Iteratio) No-liear Equatio Goal: Covergig series Rewrite Problem Example Example: Cube root % f(x) = x^3 - a = 0 % g(x) = x + C*(x^3 - a) a=2; cube.m =10; g=1.0; C=-0.1; sq(1)=g; for i=2: sq(i)= sq(i-1) + C*(sq(i-1)^3 -a); ed hold off plot([0 ],[a^(1./3.) a^(1/3.)],'b') hold o plot(sq,'r') plot( (sq-a^(1./3.))/(a^(1./3.)),'g') grid o Iteratio Numerical Fluid Mechaics PFJL Lecture 3, 23 23

24 Roots of Noliear Equatios: Bracketig vs. Ope Methods i Graphics f(x) f(x) x i xi+1 x x l x u x f(x) x l x u x l x u x i xi+1 x x l x u x l x u Image by MIT OpeCourseWare. A graphical depictio of the fudametal differeces betwee the bracketig method (show o the right) ad the ope methods (show o the left). O the right, the root is costraied withi the iterval; o the left, a formula is used to project iteratively betwee x i ad x i+1 ad ca diverge or coverge rapidly. Numerical Fluid Mechaics PFJL Lecture 3, 24 24

25 Ope Methods (Fixed Poit Iteratio) Stop-criteria Urealistic stop-criteria: cotiue while Realistic stop-criteria Machie Epsilo Use a or combiatio of the two criteria f(x) flat f(x) f(x) steep f(x) x x Caot require Caot require Numerical Fluid Mechaics PFJL Lecture 3, 25 25

26 MIT OpeCourseWare Numerical Fluid Mechaics Fall 2011 For iformatio about citig these materials or our Terms of Use, visit:

THE SOLUTION OF NONLINEAR EQUATIONS f( x ) = 0.

THE SOLUTION OF NONLINEAR EQUATIONS f( ) = 0. Noliear Equatio Solvers Bracketig. Graphical. Aalytical Ope Methods Bisectio False Positio (Regula-Falsi) Fied poit iteratio Newto Raphso Secat The root of